# Algebra II : Circle Functions

## Example Questions

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### Example Question #1 : Circle Functions

Find the -intercepts for the circle given by the equation:

Explanation:

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain .

### Example Question #1 : Circle Functions

Find the -intercepts for the circle given by the equation:

Explanation:

To find the -intercepts (where the graph crosses the -axis), we must set . This gives us the equation:

Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:

and

We can then solve these two equations to obtain

### Example Question #2 : Circle Functions

What is the greatest possible value of the -coordinate?

Explanation:

This equation describes a circle of radius  (square root of ), centered at the point .  The equation (which is NOT a function) has a maximum y-coordinate value directly above the center of the circle in the vertical direction.  Take the y-coordinate of the center, , and add to it the length of the radius, , to get the answer, .

### Example Question #3 : Circle Functions

Find the  intercept of a circle.

Explanation:

Let

Therefore the equation becomes,

Solve for x.

### Example Question #4 : Circle Functions

Find the  intercept of a circle.

Explanation:

Let

Therefore, the equation becomes:

Solve for y.

### Example Question #5 : Circle Functions

A circle centered at  has a radius of  units.

What is the equation of the circle?

Explanation:

The equation for a circle centered at the point (h, k) with radius r units is

.

Setting , , and  yields

.

### Example Question #6 : Circle Functions

Convert the following angle to radians

Explanation:

To convert degrees to radians, multiply degrees by:

Therefore

### Example Question #7 : Circle Functions

Find the negative coterminal of 160.

Explanation:

Coterminal angles are angles that are the same but written differently. Circles have 360 degrees, so an angle that goes above this threshold has completed one revolution. For example, a 450 degree angle would be in the same position as a 90 degree angle.

To find a positive coterminal angle, add 360 degrees to the initial value.

To find the negative coterminal angle, simply subtract 360. The only exception to this rule would be if the initial value were greater than 360. In this case, subtract 360 until the value is negative, making it a negative coterminal. Therefore,

160-360 = -200

Do this a second time and we get -560. These are examples of negative coterminal angles.

### Example Question #8 : Circle Functions

Which of the following equations represent a circle?

Explanation:

The circle is represented by the formula:

Although some of the equations might not in this form, we can see by the variables that the equation   is most similar to the form.

Multiply two on both sides of the equation and we will have:

This is an equation of a circle.  The other equations represent other conic shapes.

### Example Question #9 : Circle Functions

Determine the equation of a circle that has radius and is centered at

None of these

Explanation:

Definition of the formula of a circle:

Where:

is the coordinate of the center of the circle

is the  coordinate of the center of the circle

is the radius of the circle

Plugging in values:

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